Compression body

Template:One source In the theory of 3-manifolds, a compression body is a kind of generalized handlebody.

A compression body is either a handlebody or the result of the following construction:

Let $S$ be a compact, closed surface (not necessarily connected). Attach 1-handles to $S\times [0,1]$ along $S\times \{1\}$.

Let $C$ be a compression body. The negative boundary of C, denoted ${\partial }_{-}C$, is $S\times \{0\}$. (If $C$ is a handlebody then ${\partial }_{-}C=\mathrm{\varnothing }$.) The positive boundary of C, denoted ${\partial }_{+}C$, is $\partial C$ minus the negative boundary.

There is a dual construction of compression bodies starting with a surface $S$ and attaching 2-handles to $S\times \{0\}$. In this case ${\partial }_{+}C$ is $S\times \{1\}$, and ${\partial }_{-}C$ is $\partial C$ minus the positive boundary.

Compression bodies often arise when manipulating Heegaard splittings.

References

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